Theorem dfrnf 4973

Description: Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
dfrnf.1	⊢ Ⅎ𝑥𝐴
dfrnf.2	⊢ Ⅎ𝑦𝐴

Assertion

Ref	Expression
dfrnf	⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem dfrnf

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrn2 4918	. 2 ⊢ ran 𝐴 = {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤}
2		nfcv 2374	. . . . 5 ⊢ Ⅎ𝑥𝑣
3		dfrnf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2374	. . . . 5 ⊢ Ⅎ𝑥𝑤
5	2, 3, 4	nfbr 4135	. . . 4 ⊢ Ⅎ𝑥 𝑣𝐴𝑤
6		nfv 1576	. . . 4 ⊢ Ⅎ𝑣 𝑥𝐴𝑤
7		breq1 4091	. . . 4 ⊢ (𝑣 = 𝑥 → (𝑣𝐴𝑤 ↔ 𝑥𝐴𝑤))
8	5, 6, 7	cbvex 1804	. . 3 ⊢ (∃𝑣 𝑣𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑤)
9	8	abbii 2347	. 2 ⊢ {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} = {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤}
10		nfcv 2374	. . . . 5 ⊢ Ⅎ𝑦𝑥
11		dfrnf.2	. . . . 5 ⊢ Ⅎ𝑦𝐴
12		nfcv 2374	. . . . 5 ⊢ Ⅎ𝑦𝑤
13	10, 11, 12	nfbr 4135	. . . 4 ⊢ Ⅎ𝑦 𝑥𝐴𝑤
14	13	nfex 1685	. . 3 ⊢ Ⅎ𝑦∃𝑥 𝑥𝐴𝑤
15		nfv 1576	. . 3 ⊢ Ⅎ𝑤∃𝑥 𝑥𝐴𝑦
16		breq2 4092	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑥𝐴𝑤 ↔ 𝑥𝐴𝑦))
17	16	exbidv 1873	. . 3 ⊢ (𝑤 = 𝑦 → (∃𝑥 𝑥𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑦))
18	14, 15, 17	cbvab 2355	. 2 ⊢ {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
19	1, 9, 18	3eqtri 2256	1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Syntax hints:   = wceq 1397  ∃wex 1540  {cab 2217  Ⅎwnfc 2361   class class class wbr 4088  ran crn 4726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by:  rnopab  4979